Abstract

Here a modification to the hybrid input–output (HIO) method of phase retrieval is presented which aides in mitigating the negative effects of low signal-to-noise ratios (SNRs). Various type of interferometers measure diffraction patterns which are used to determine the Fourier transform modulus of an objective. Interferometry often suffers from very low SNRs making phase retrieval difficult because of the sensitivity of most phase retrieval algorithms to local minima. Here we analyze the effect of noise on the HIO method. The result is used as a rationale for the proposed modification to the HIO method. The algorithm presented here introduces a filtering scheme which removes much of the Fourier modulus noise. Examples are shown and the results are compared to the HIO method with and without the proposed modification. Comparisons are also made to other methods of filtering the Fourier modulus noise.

© 2013 Optical Society of America

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef]
  3. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  4. R. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
    [CrossRef]
  5. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
    [CrossRef]
  6. G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Signal Process. 21, 339–347 (1990).
    [CrossRef]
  7. G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
    [CrossRef]
  8. R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
    [CrossRef]
  9. M. Kohl, A. A. Minkevich, and T. Baumback, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express 20, 17093–17106 (2012).
    [CrossRef]
  10. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  11. D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
    [CrossRef]
  12. R. H. Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
    [CrossRef]
  13. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  14. A. V. Oppenhein and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Prentice-Hall, 1999).
  15. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78011301 (2007).
    [CrossRef]
  16. A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  17. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
  18. I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
    [CrossRef]
  19. J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
    [CrossRef]
  20. J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004).
    [CrossRef]
  21. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]

2012 (2)

M. Kohl, A. A. Minkevich, and T. Baumback, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express 20, 17093–17106 (2012).
[CrossRef]

D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
[CrossRef]

2011 (1)

J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
[CrossRef]

2007 (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78011301 (2007).
[CrossRef]

2004 (1)

2003 (2)

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

1998 (1)

1996 (1)

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

1990 (2)

G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Signal Process. 21, 339–347 (1990).
[CrossRef]

G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
[CrossRef]

1986 (2)

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

1984 (1)

1982 (2)

R. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1956 (1)

R. H. Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[CrossRef]

Bates, R.

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

R. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

Baumback, T.

Brown, R. H.

R. H. Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Chapman, H. N.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Dravins, D.

D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
[CrossRef]

Elser, V.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

He, H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Honda, T.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Ito, A.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Jensen, H.

D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
[CrossRef]

Kinoshita, K.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Kodama, I.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Kohl, M.

LeBohec, S.

D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
[CrossRef]

Levi, A.

Liu, G.

G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Signal Process. 21, 339–347 (1990).
[CrossRef]

G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
[CrossRef]

Marchesini, S.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78011301 (2007).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Matsumura, T.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Miao, J.

Minkevich, A. A.

Mnyama, D.

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Nunez, P.

D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
[CrossRef]

Ohyama, N.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Oppenhein, A. V.

A. V. Oppenhein and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Prentice-Hall, 1999).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sayre, D.

Schafer, R. W.

A. V. Oppenhein and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Prentice-Hall, 1999).

Shinohara, K.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Spence, J.

J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Stark, H.

Twiss, R. Q.

R. H. Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[CrossRef]

Wackerman, C. C.

Wang, D.

J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
[CrossRef]

Wang, Y.

J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
[CrossRef]

Weierstall, U.

J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Wu, J. S.

Yada, K.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Yamaguchi, M.

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

Zhang, F.

J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
[CrossRef]

Zhao, J.

J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
[CrossRef]

Adv. Electron. Electron Phys. (1)

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

Appl. Opt. (1)

Astropart. Phys. (1)

D. Dravins, S. LeBohec, H. Jensen, and P. Nunez, “Optical intensity interferometry with the Cherenkov telescope array,” Astropart. Phys. 43, 331–347 (2012).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nature (1)

R. H. Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[CrossRef]

Opt. Commun. (2)

I. Kodama, M. Yamaguchi, N. Ohyama, T. Honda, K. Shinohara, A. Ito, T. Matsumura, K. Kinoshita, and K. Yada, “Image reconstruction from an in-line X-ray hologram,” Opt. Commun. 125, 36–42 (1996).
[CrossRef]

G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
[CrossRef]

Opt. Eng. (1)

J. Zhao, D. Wang, F. Zhang, and Y. Wang, “Hybrid phase retrieval approach for reconstruction of in-line digital holograms without twin image,” Opt. Eng. 50, 091310 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rep. (1)

R. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78011301 (2007).
[CrossRef]

Signal Process. (1)

G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Signal Process. 21, 339–347 (1990).
[CrossRef]

Other (1)

A. V. Oppenhein and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Prentice-Hall, 1999).

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Figures (9)

Fig. 1.
Fig. 1.

Example result used to demonstrate constraint oscillation in the HIO. The box indicates the boundary of γ.

Fig. 2.
Fig. 2.

Image constraint violation versus iteration for the HIO.

Fig. 3.
Fig. 3.

Modulus constraint violation versus iteration for the HIO.

Fig. 4.
Fig. 4.

Image estimate comparison before and after the constraint relaxation was implemented. The box indicates the boundary of γ.

Fig. 5.
Fig. 5.

Image constraint violation versus iteration. Constraint relaxation began at iteration 500.

Fig. 6.
Fig. 6.

Modulus constraint violation versus iteration. Constraint relaxation began at iteration 500.

Fig. 7.
Fig. 7.

Fourier modulus error versus iteration.

Fig. 8.
Fig. 8.

Sum of the image’s imaginary pixels prior to the image constraint being imposed versus iteration.

Fig. 9.
Fig. 9.

Example result showing (a) the true image and (b) the reconstructed image after 500 iterations without constraint relaxation. The relaxation was performed from iteration 500 to 1000 with the result shown in (c). The box in (b) and (c) indicates the boundary of γ.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

f(x,y)=F1{|F(u,v)|exp(iφ(u,v))},
Gk(u,v)=|F(u,v)|exp(iarg(Gk(u,v))).
gk+1(x,y)={gk(x,y),(x,y)γgk(x,y)βgk(x,y),(x,y)γ,
|F(u,v)|=|F(u,v)|=|FTrue(u,v)(1+N[0,σ]+N[0,σ]i)|
E˜k2=N2(|Ftrue(u,v)||Gk(u,v)|)2,
ek2=γgk(x,y)2,
Ek2=N2(|Gk(u,v)|F(u,v))2,
G˜(u,v)=FTrue(u,v)+(a,b)Δ(a,b)δ(ua,vb).
G(u,v)=Δδ(ua,vb).
f(x,y)=ΔMNe2πi((ax/M)+(by/N)),
f(x,y)={|f(x,y)|,AmM1A&BnN1B0,otherwise=ΔMN{1,AmM1A&BnN1B0,otherwise.
G(u,v)=ΔMNexp(iπ(uM4A1M+vN4B1N))sin(M2AMπu)sin(1Mπu)sin(N2BNπv)sin(1Nπv).
G(0,0)=ΔM2AMN2BN.
M2AMN2BN.
ΔM2AMN2BN,
|Gk(u,v)|=(1λ)F(u,v)+λ|Gk(u,v)|.
|Gk(u,v)|=(1λ)|Gk1(u,v)|+λ|Gk(u,v)|,
Gk(u,v)=(1λA)Gk(u,v)+λAF(u,v)exp(iarg(Gk(u,v)))
arg(gk(u,v))2

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